I've got the Bulbank app too for my BG account, it's OK.
I also have Revolut, which is one of the more popular international banking apps. I think a lot of expats have this. Good app, multicurrency, euro debit card, virtual cards, direct debits. I think one could use it instead of a local (BG or wherever) account, but I haven't committed to it sufficiently to try this. Similar would be N26, also very popular across Europe.
PayPal & Wise emoney, I use Wise a fair bit. But no apps (that I've seen anyway).
Just FYI: I think the fairly official terminology is "ebank" or "neobank" for online biz with real banking license, while "emoney" is similar, but without banking (but maybe have other type) license. So Starling, Monzo, N26 are ebanks. While Wise, PayPal, Revolut are emoney. Some of these new fintech outfits are pretty big now, so safety shouldn't be an issue. But, technically, protection for your funds provided by two different mechanisms (ebank, covered by national banking protection and emoney by "segregation" and insurance policy). These outfits might be app or web (or both), but don't have physical branches like the old school stuff. (Instead, they have access to ATM network for withdrawing cash, and maybe some specific arrangement with existing physical bank/post offices for paying in cash.)
I see your heading is "keeping money safe", which suggests you have a lot of it, lucky you! :-) If it's a few hundred or a few thousand (or your monthly salary deposit), I'd avoid the new Joe Bloggs Awesome Banking App, but any of these big names will be pretty safe for you. If you're thinking tens or hundreds of thousands, then I'd look more carefully at ebanks, and what the national banking protection covers (e.g. EU-wide deposit protection scheme up to 100,000 euros). Separately, deposit / savings interest rates are negative in Bulgaria (and many other countries), so while your money might be safe, you will be paying the bank for holding on to it, rather than them paying you. Hence, in Bulgaria, for instance, big rush to move savings into property thus fuelling house price inflation.